1. ## [SOLVED] Intgers

If a and b are positive integers and $\displaystyle (a^{1/2}b^{1/3})^6=432$, what is the value of ab?

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2. Originally Posted by fabxx
If a and b are positive integers and $\displaystyle (a^{1/2}b^{1/3})^6=432$, what is the value of ab?

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note that you have $\displaystyle a^3b^2 = 432$

so you want to express 432 as the product of a square and a cube. find the prime factorization of 432, that should help

3. Originally Posted by Jhevon
note that you have $\displaystyle a^3b^2 = 432$

so you want to express 432 as the product of a square and a cube. find the prime factorization of 432, that should help
you mean $\displaystyle 3^3$ x $\displaystyle 2^4$?

4. Originally Posted by fabxx
you mean $\displaystyle 3^3$ x $\displaystyle 2^4$?
yes

can you express that as something cubed times something squared?

5. How about 4^2 times 3^3? it's something squared and something cubed

6. Originally Posted by fabxx
How about 4^2 times 3^3? it's something squared and something cubed
yes. so what is your a and b? and hence, ab

7. thank you. so the final answer should be 4x3=12

I have a question from your earlier post:

Originally Posted by Jhevon
note that you have $\displaystyle a^3b^2 = 432$
how did you get from $\displaystyle a^\frac{1}{2}b^\frac{1}{3}=432$ to $\displaystyle a^3b^2 = 432$?? and also what about the 6 from [a^(1/2)b^(1/3)]^6? How about the 6 here? Thanks again!!

8. Originally Posted by fabxx
thank you. so the final answer should be 4x3=12
yes

I have a question from your earlier post:

how did you get from $\displaystyle a^\frac{1}{2}b^\frac{1}{3}=432$ to $\displaystyle a^3b^2 = 432$?? and also what about the 6 from [a^(1/2)b^(1/3)]^6? How about the 6 here? Thanks again!!
here is the rule: $\displaystyle (x^a)^b = x^{ab}$ and also $\displaystyle (x^ay^b)^c = x^{ac}y^{bc}$

that is, we can distribute the power and multiply them when we raise a number to a power to another power